What kind of bite or sting would cause a red welt surrounded by a pink ring a week after the bite?

Tick, spider: It is hard to say what bit you or where you were when you got bitten. A tick bite will have a red ring like a bulls eye around it. You may have mild flu like symptoms that could be lyme disease.A spider bite can also leve a red welt with a ring. Sensuitivity to many insect bites can do this. If you have concerns and it is not healing then you should see a doctor and get it checked.

Many possibilities: This rash needs to be visualized & a history needs to be obtained in order to make an accurate diagnosis.